For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

learn more… | top users | synonyms (2)

2
votes
4answers
123 views

Show that $AB=0 \iff A=A^2$ and $B=B^2$

Let $n$ be a positive integer and let $\Bbb F$ be a field. Let $A,B \in M_{n\times n} (\Bbb F)$ satisfy $A+B=I$. Show that $AB=0 \iff A=A^2$ and $B=B^2$ where $0$ is the zero matrix. If $A,B$ are ...
1
vote
2answers
56 views

Find a nonzero matrix $A$ in $M_{2\times 2}(R)$ satisfying $v\cdot Av=0$ for every $v\in R^2$

Ok, so I have this problem: Find a nonzero matrix $A$ in $M_{2\times 2}(R)$ satisfying $v\cdot Av=0$ for every $v \in R^2$. So if I say that $v=\begin{bmatrix}x\\y\end{bmatrix}$ and that ...
2
votes
0answers
31 views

Cayley on “trivial transformations”

In his 1854 paper, "Deuxième mémoire sur les fonctions doublement périodiques" ("Second memoir on doubly periodic functions"), Cayley discusses (what we would today describe as) a certain class of ...
2
votes
3answers
77 views

$AA^{t}=A^{t}A$ in this case?

Let $M$ be a square matrix such that $M_{i,j}=c$ if $i\not=j$ and $d$ if $i=j$. Also, $0<c<d$ (note that this is a strict inequality). That is, $M$ look like this: $\begin{pmatrix}d & c ...
10
votes
3answers
231 views

Existence of some type matrix

Is there square matrix $A$ of size $3$ with real entries such that $$ \operatorname{tr}(A)=0\text{ and }A^2+A^T=I. $$ I have proved that there is not with size $2$ using definition of "trace", but ...
0
votes
1answer
22 views

Question about Joint spectral radius.

Given a bounded set $\mathcal A\subset \Bbb R^{n x n}$. The joint spectral radius is given by: $\sigma(\mathcal A)$=$limsup_{m\to\infty}(sup_{A\in\mathcal A^m} \rho(A))$ where $\rho$ is the normal ...
2
votes
2answers
974 views

Mathematical Induction Matrix Example

I'm a little rusty and I've never done a mathematical induction problem with matrices so I'm needing a little help in setting this problem up. Show that ...
0
votes
1answer
303 views

All possible combinations from a matrix

I would like to get all possible combinations from a matrix as follows: $$ A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{bmatrix}, ...
1
vote
2answers
50 views

Let L be a linear transformation defined by its standard matrix AL

$\begin{bmatrix}1&-1&1&2\\2&-2&3&4\end{bmatrix}$ As I understand domain and codomain of L are $L:\mathbb{R}^4 \rightarrow \mathbb{R}^2$ ? How can I write formula definition ...
1
vote
1answer
28 views

Vectors, columns and representations

When I learned quantum mechanics, my professor frequently emphasized that a matrix is a representation of an operator, not the operator itself, and a column ($1\times M$ matrix) is not a vector, it's ...
1
vote
1answer
124 views

Diagonalization of a Toeplitz matrix

Let $0<\lambda\leq1$ so that the $n \times n$ matrix $$\Sigma = \begin{pmatrix} 1&1-\lambda& \cdots &1-\lambda\\ 1-\lambda&\ddots&\ddots& \vdots\\ \vdots ...
1
vote
2answers
36 views

Find all 2-order orthogonal matrices

Actually, there are only two kinds of 2-order orthogonal matrices: $$\left( \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{array} \right)$$ or $$\left( ...
1
vote
1answer
67 views

Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
5
votes
6answers
703 views

Can an empty array be useful?

Most computer programming languages have constructs for managing arrays of data, including multiple-dimensional arrays, which are clearly useful when storing, manipulating and modelling mathematical ...
1
vote
1answer
103 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
1
vote
1answer
39 views

Diagonalization of a Toeplitz matrix

Let $0<\lambda\leq1$ so that the $n \times n$ matrix $$\Sigma = \begin{pmatrix} 1&1-\lambda& \cdots &1-\lambda\\ 1-\lambda&\ddots&\ddots& \vdots\\ \vdots ...
0
votes
1answer
22 views

Question regarding trasposes and norms

I was pondering my book of linear algebra and I found this solution to question 3.2.2 here; but the author of the solution follows a path that I am not sure it is correct... when he takes the ...
1
vote
1answer
121 views

Prove that $Im(A)+Ker(A)=R^n \iff Ker(A^2)=Ker(A)$

$\def\Im{\operatorname{Im}}\def\Ker{\operatorname{Ker}}$How to prove that for any squared matrix such that $ \Im(A)+\Ker(A)=\mathbb{R}^n$ if and only if $\Ker(A^2)=\Ker(A)$. It is evident to me that ...
0
votes
1answer
40 views

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$?

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$? For example, Can I say that $M^R_{2x2}$ is a subspace of $M^R_{2x3}$ so it can be isomorphic to $R_4[x]$ ? (because they have ...
0
votes
1answer
253 views

Notation for matrix and sum of matrix rows

I have a table that describes the influence of sources (columns) on sinks (rows) where rows=$(A,B,C)$ and columns=$(A,B,C,D,E)$. So my table looks like: ...
3
votes
1answer
66 views

Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)

Let $\mathbb K$ denote the skew field of quaternions and $A \in M^{n \times n}(\mathbb K)$ and $X\in M^{1\times n}(\mathbb K)$. Let $L_A : \mathbb K^n \to \mathbb K^n$ be defined as $L_A(X) = ...
4
votes
2answers
95 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
3
votes
2answers
119 views

Group does not contain any elements of order $p^2$?

I am trying to show that the group of $3 \times 3 $ upper triangular matrices over the field $ \mathbb{F}_p $ with diagonal entries 1 does not contain any elements of order $p^2$ when $ p \geq 3$. ...
0
votes
1answer
36 views

Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
0
votes
3answers
126 views

Finding ker, im, dim of a linear transformation

1Ok, I am a student trying to wrap my head around some of these concepts and need help understanding how to approach some problems. Question: Let $\alpha:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the ...
1
vote
0answers
36 views

Computing Integrals over space of Hermitian Matrices.

I am currently working on an example for my research, and I'm getting stuck. Without going over the details of what things mean explicitly, I would like some help computing this integral. I will ...
5
votes
1answer
181 views

When is the equation $Ax = b$ solvable in the integers?

Let $A$ be an $m\times n$ matrix with integer entries, $b$ a column-vector with $m$ integer entries. Suppose the equation $Ax = b$ has infinitely many solutions. It is clear that the general ...
1
vote
1answer
162 views

can someone help me to prove rank(P A) = rank(A).?

is that correct and we should use the hint but how we use it correctly??
1
vote
0answers
36 views

Matrix almost similar to identity?

I wonder if I can find for any matrix $B\in \mathbb{R}^{n\times n}$ a regular matrix $A\in \mathbb{R}^{n\times n}$ that minimises $$ || A^{-1} B A - Id ||_F $$ where $||.||_F$ denotes the frobenius ...
1
vote
1answer
234 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
5
votes
1answer
70 views

Eigenvalues of a $4\times 4$ parameters matrix

Let $a,b,c,d\in\Bbb{C}$ and $B =\begin{bmatrix} a & b & c & d\\ d & a & b & c\\ c & d & a & b\\ b & c & d & a\\ \end{bmatrix}$ I ...
1
vote
0answers
128 views

what does autocorrelation matrix signifies in image processing?

I am trying to find out corners using auto correlation matrix in an image,what does it signifies?
0
votes
1answer
38 views

Constructing an eigenvector for a certain matrix representing a graph with a perfect code

Let $A$ be a symmetric $(0,1)$-matrix whose row sum is $r.$ Suppose I have a $(0,1)$ vector $v$ such that $$Av = \vec{1} - v.$$ By taking the vector $$u = \vec{1} - (r+1)v$$ we see that $$Au = A ...
0
votes
0answers
39 views

Find characteristic polynomial

Suppose $A$ and $B$ are $n \times n$ complex matrices such that $$AB-BA=aI+A,$$ where $a \in \mathbb{C}.$ Find the characteristic polynomial of $A$. If $A$ happens to be a Jordan block, this would be ...
0
votes
1answer
19 views

Matrix norm can assume infinite values?

Given a real-valued matrix $A=a_{i,j} \in M_{n,n}$ When: $||A||_2= \sqrt {\sum_{i=0}^n a^2_{i,j}} < + \infty$ ? Why?
3
votes
0answers
38 views

what is the significance of Eigen values of autocorrelation matrix?

I am trying to find auto correlation matrix of an image to get Harris corners.Paper I am referring suggest that if eigen values of auto correlation matrix are large the point will be corner point.so ...
1
vote
1answer
55 views

Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$ Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant. How can write it as in roots ...
0
votes
1answer
40 views

Finding the canonical form of a matrix

$$A= \begin{bmatrix} 2 & 0 & -1 \\ -5 & 3 & 3 \\ \end{bmatrix}$$ I have to find two invertible matrices $P(2\times 2)$, $Q(3\times 3)$ such that $P^TAQ$ is a canonical matrix. I know ...
3
votes
2answers
200 views

Linear algebra calculus trick.

I have a matrix and a vector: $$ A=\begin{bmatrix} a &b\\ c&d \end{bmatrix}, $$ $$ \vec v=\begin{bmatrix} a+b\\ c+d \end{bmatrix} $$ Is there an algebraic operation that produce the ...
1
vote
1answer
41 views

Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
2
votes
2answers
158 views

Hadamard Matrix

Prove that if $H$ is a (normalized) Hadamard matrix, then so is the matrix $\pmatrix{ H& H\\\ H& -H}$. I have been working on this and I know this statement is true. My book just simply says ...
2
votes
1answer
71 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
1
vote
1answer
102 views

Why diagonal matrix SVD sorted from largest to smallest value?

Why diagonal matrix SVD sorted from largest to smallest value? D is diagonal matrix, $D=(d_1 \ge ,d_2 \ge ,..., \ge d_L)$. Whether there is a journal that could explain this?
0
votes
1answer
79 views

Matrix inverse and Change of basis

I have 2 Change of Basis Matrices $ S_{A,B} $ and $ S_{A,C}$ I want determinate $ S_{C,B} $ We know that $$ S_{A,B} S_{B,C} = S_{A,C} $$ $$ S_{B,C} = S_{A,B}^{-1} S_{A,C} $$ Now i'm quite not ...
3
votes
3answers
223 views

Matrices as generators of free group.

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing... In my words: The matrices $\begin{pmatrix}1 & 0 \\ 2 ...
1
vote
3answers
60 views

$A$ is not similar to a diagonal matrix over the reals

Let $A = \begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 & -3 \end{bmatrix} $ then $A$ is not similar to a diagonal matrix over the reals and it is not similar to a ...
0
votes
1answer
205 views

Inverse of a block 2x2 matrix

How to solve this type of problem: We've got a block 2x2 matrix : $$A=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}$$ If matrices $A$ and $A_{22}$ are invertible, show that a ...
0
votes
1answer
46 views

Vector proportional to column of cofactors

Let $A$ be a skew-symmetric $n\times n$ matrix over real numbers with rank $n-1$, and let $v$ be a vector such that $Av=0$. Let $p_{i}$ be the cofactor at position $(i,1)$. Suppose that $p_i> 0$ ...
2
votes
1answer
173 views

Principal ideal ring

Let $K$ be a principal ideal ring. How to prove that for any $ x= (x_1, x_2)^t \in K^2 $ there exists a matrix $G \in SL_2(K)$ such that $Gx = (\gcd(x_1, x_2),0)^t $ ?
4
votes
2answers
143 views

Do cyclic permutations of rows and column entries generate all permutations?

Background: I am interested in the group of permutations of the entries of a general $m\times n$ matrix. In particular, I am interested in (1) interesting sets of simple generators for this group ...